Complex-Network Modelling and Inference

Transcription

1 Complex-Network Modelling and Inference Lecture 12: Random Graphs: preferential-attachment models Matthew Roughan Network_Modelling/ School of Mathematical Sciences, University of Adelaide August 21, 2018

2 Section 1 Preferential attachment graphs August 21, / 24

3 Problem with small-world graph Small-world graph replicates desired short path length high clustering Node degree is (almost) always k but observed node-degree distributions are more variable August 21, / 24

4 Scale-free Networks Barabási and Albert [BA99] Draw on idea that the rich get richer Preferential attachment model 1 start with N = {1, 2}, and E = {(1, 2)}. 2 for i=3:n a add vertex i to N b add link (i, j) to E, where j {1, 2,..., i 1} is chosen with probability k j p j = i 1 k=1 k, k where k j is the degree of node j. August 21, / 24

5 Scale-free Networks, mark II Barabási and Albert [BA99] Draw on idea that the rich get richer Preferential attachment model 1 start with N = {1, 2}, and E = {(1, 2)}. 2 for i=3:n a add vertex i to N b add m links (i, j) to E, where j {1, 2,..., i 1} is chosen with probability k j p j = i 1 k=1 k, k where k j is the degree of node j. August 21, / 24

6 Properties of preferential attachment connected (by construction) degree distribution takes power-law form p k k α. August 21, / 24

7 Degree distribution approximation Take degree k i of ith node to be a continuous variable Take time (number of nodes added) to be continuous Rate of increase of degree is proportional to degree dk i dt = m k i n j=1 k j note that the total number of links in the network is n E = mt = k j /2 j=1 August 21, / 24

8 Degree distribution approximation Substitute 2mt in first equations Solve the DE, and we get dk i dt = k i 2t. Use initial condition k i (t i ) = m k i (t) = ct 1/2 k i (t) = m(t/t i ) 1/2 August 21, / 24

9 Degree distribution approximation So Calculating the CDF we get k i (t) = m(t/t i ) 1/2 Prob{k i (t) < k} = Prob{m(t/t i ) 1/2 < k} = Prob{(t/t i ) < (k/m) 2 } = Prob{(t i /t) > (m/k) 2 } = 1 Prob{(t i /t) (m/k) 2 } Adding nodes at uniform time intervals means t i = i, so in the limit as t, the t i /t are uniformly distributed on [0, 1], and we get the form Prob{k i (t) < k} 1 (m/k) 2 for (m/k) 2 1 August 21, / 24

10 Degree distribution approximation For large k, (m/k) 2 1, and the density function p k can be approximated by the derivative p k d dk Prob{k i(t) < k} d dk (m/k)2 2m 2 k 3 we usually care about the limit (for this type of distribution) so we write p k k 3 This is a power law with exponent 3 August 21, / 24

11 Generalisation Evolve the network over time add m edges with probability p one end uniformly chosen over all nodes other end chosen proportional to degree rewire m edges with probability q choose node i at random rewire one of its edges using proportional attachment with probability 1 p q a new node is added m new edges with proportional attachment Can generate degree distribution with power-law between 2 and. August 21, / 24

12 Why Scale Free degree distribution doesn t depend on the size of the network (as long as its a limit) form of degree distribution doesn t depend on number of links (per node) power-laws exhibit a type of scale invariance p(x) = ax α another form of scale invariance regardless of x p(bx) = a(bx) α = Ax α p(x) p(2x) = 2 α p(x) August 21, / 24

13 Power-laws Power-laws look like straight lines on a log-log graph CCDF, Pr{k i k} degree August 21, / 24

14 Do they match real data? Actor collaboration graph appears to have power-law [BA99] PDF actor degree August 21, / 24

15 Do they match real data? Care must be taken though CCDF actor 10 3 degree August 21, / 24

16 Do they match real data? AS-graph appears to have power-law [FFF99] PDF AS in degree August 21, / 24

17 Do they match real data? Care must be taken though CCDF AS in degree August 21, / 24

18 PDF vs CCDF for continuous distributions PDF = derivative of CDF = - derivative of CCDF if one has a power-law, both should PDF: Prob{k i == k} hard to accurately estimate require arbitrary choice of binning lots of zeros in the tail zeros don t show up on log-log graph CCDF: Prob{k i > k} easy to estimate/plot loglog(sort(degree), 1 - (0:n-1)/n) much more robust in the tail August 21, / 24

19 Do they match real data? WWW page graph really appears to have power-law [AJB99] CCDF www page in degree August 21, / 24

20 Do they match real data? WWW page graph really appears to have power-law [AJB99] CCDF www page out degree August 21, / 24

21 Power-law degree Appeal of the model simple/parsimonious real networks sometimes have power-law degree makes more sense for virtual networks power-law is an emergent phenomena seems logical, but wait Even if they match data, does the model explain the real process behind network construction is this the only way to generate power-laws? if not, does the model tell us anything? do other features match real networks? We ll come back to these topics after we consider measurements in more detail. August 21, / 24

22 Preferential attachment generalisations Price s model: We can make the number of edges brought by a new node random We can allow some re-wiring allows varying power-law exponent Can allow node birth and death of nodes August 21, / 24

23 Estimation In BA model, it comes down to estimating average degree In general, need to estimate exponent of a power-law more on this later August 21, / 24

24 Further reading I R eka Albert, Hawoong Jeong, and Albert-László Barabási, Diameter of the world wide web, Nature 401 (1999), no. 130, A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science 286 (1999), Michalis Faloutsos, Petros Faloutsos, and Christos Faloutsos, On power-law relationships of the Internet topology, ACM SIGCOMM 99, August 21, / 24